A characterization of consistent estimators

Strong consistency results have been established for maximum likelihood estimates (MLE´s), least square estimates (LSE´s), and more recently for prediction error estimates (PEE´s). The basic characteristic of each of these estimates is that they are defined in terms of extremum values of some appropriate function of the observed data and the unknown parameters. The strong consistency results that are presently available require conditions on the appropriate functions that include MLE´s, LSE´s, and PEE´s, respectively. Conditions such as differentiability with respect to the unknown parameters, existence of certain limits, availability for a certain type of systems, etc., are usually required. In this paper we will present a reasonably general characterization of strong consistency which apparently allows us to treat a broader class of estimation problems than has been treated before. We establish that strong consistency is basically a question of limits of the extremal points of a suitable sequence of functions of the observations. This sequence of functions must satisfy certain almost sure asymptotic properties; otherwise they are quite arbitrary.